Calculus PYQ — Page 5

Consider the differential equation $xdy = (x + y) dx$. Which of the following are true? (A) It is a homogenous differential equation (B) It is a differential equation of order 2 (C) The general solution of the differential equation contains 2 arbitrary constants (D) Integrating factor of differential equation is $\frac{1}{x}$ (E) Degree of the differential equation is not defined Choose the correct answer from the options given below:

The equation of tangent line to $y = 2x^2 + 7$, which is parallel to the line $4x - y + 3 = 0$ is

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equation | Order and degree of differential equation | | (A) $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1$ | (I) Order = 1, Degree = 2 | | (B) $\left(\frac{d^2y}{dx^2}\right)^2 + 4\left(\frac{dy}{dx}\right)^3 = e^y - 1$ | (II) Order = 2, Degree = 1 | | (C) $3\left(\frac{dy}{dx}\right) + 4y + e^y = \frac{dx}{dy}$ | (III) Order = 2, Degree = 2 | | (D) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right) = \left(e^y + \frac{dy}{dx}\right)^2$ | (IV) Order = 2, Degree = Not defined | Choose the correct answer from the options given below:

The solution of the differential equation $\frac{dy}{dx} = \frac{x+y}{x-y}$ is

If $y = x\sin y$, then $\frac{dy}{dx}$ is:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The value of $\int_{0}^{4} \vert x\vert \, dx$ is | (I) 3 | | (B) The value of $\int_{-2}^{2} \vert x\vert \, dx$ is | (II) -1 | | (C) The value of $\int_{0}^{3} [x] \, dx$ is | (III) 8 | | (D) The value of $\int_{-1}^{1} [x] \, dx$ is | (IV) 4 | Choose the correct answer from the options given below:

The area of the region bounded by parabola $x^2 = 4y$, straight line $x = 2$ and $x$-axis, is

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is

Match List-I with List-II | List-I | List-II | |---|---| | (Differential equation) | (Order and Degree) | | (A) $\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0$ | (I) order = 3, degree = 1 | | (B) $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} + 1 = 0$ | (II) order = 3, degree not defined | | (C) $2x^2\frac{d^2y}{dx^2} - 3\left(\frac{dy}{dx}\right)^2 + y = 0$ | (III) order = 2, degree = 3 | | (D) $\frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0$ | (IV) order = 2, degree = 1 | Choose the correct answer from the options given below:

For $x \neq -1$, if $\int \frac{xe^x dx}{(1+x)^2} = \frac{ae^x}{(1+x)^b} + c$, where a, b are fixed numbers and c is the integration constant, then $a + b$ is equal to

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$. If it passes through the point $(0,0)$, then which of the following is/are true? (A) it is a straight line. (B) it is a parabola. (C) it also passes through the point $(-1, \frac{1}{\sqrt{e}} - 1)$ (D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$ Choose the **correct** answer from the options given below:

Which of the following are correct? (A) The function $f(x) = 3x+12$ is increasing on R. (B) The function $f(x) = e^{2x}$ is decreasing on R. (C) The function $f(x) = x^2-x-1$ is neither increasing nor decreasing on (-1, 1). (D) The function $f(x) = x^3-3x^2+4x$ is increasing on R. Choose the correct answer from the options given below:

If $\int e^x\left(\frac{x-1}{(x+1)^3}\right)dx = \frac{Ae^x}{(x+1)^B} + C$, where C is constant of integration then which of the following are correct? (A) $A = -1$ (B) $A = 1$ (C) $B = 3$ (D) $B = 2$ Choose the correct answer from the options given below:

Which of the following functions are increasing on $x \in \left(0, \frac{\pi}{2}\right)$? (A) $f(x) = \sin x$ (B) $f(x) = \cos x$ (C) $f(x) = \tan x$ (D) $f(x) = \cos 3x$ Choose the correct answer from the options given below:

The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is

Match List-I with List-II | List-I | List-II | | :--- | :--- | | **Differential equation** | **Order and degree** | | (A) $(y'')^3 + (y')^4 - 6 = (y''')^2$ | (I) Order = 1, Degree = 2 | | (B) $\sqrt{(y')^2 + 5} = y''$ | (II) Order = 2, Degree = 3 | | (C) $(y')^2 = (2 + y'')^{3/2}$ | (III) Order = 2, Degree = 2 | | (D) $y = xy' + \sqrt{a^2(y')^2 + b^2}$ | (IV) Order = 3, Degree = 2 | Choose the correct answer from the options given below:

The value of the definite integral $I = \int_{-1}^{1} \frac{1}{1 + \sqrt{e^x}} dx$ is:

If $e^y(x + 1) = 1$, then

If $x = a\cos\alpha + b\sin\alpha$ and $y = a\sin\alpha - b\cos\alpha$, then $\left(x\frac{dy}{dx} - y^2\frac{d^2y}{dx^2}\right)$ is equal to:

Match List-I with List-II [.] denotes the greatest integer function. | List-I | List-II | |---|---| | (A) $\int_0^3 [x]dx$ | (I) $\frac{1}{2}$ | | (B) $\int_0^1 [2x]dx$ | (II) 1 | | (C) $\int_0^1 [3x]dx$ | (III) $\frac{3}{2}$ | | (D) $\int_0^1 [4x]dx$ | (IV) 3 | Choose the correct answer from the options given below:

The general solution of the differential equation $\frac{xdy}{dx} + 4y = x^3, (x \neq 0)$ is:

The interval on which the function $f(x) = x^4 - \frac{x^3}{3}$ is strictly decreasing, is:

For $x > 1$, $\int \frac{e^{7\log x} - e^{5\log x}}{e^{5\log x} - e^{4\log x}} dx$ equals.

$\int \frac{(x-3)e^x}{(x-1)^3} dx$ is equal to